How do I make a curve that goes through a series of points?
You can &quot;draw&quot; connected points by pressing Ctrl and the right mouse button. The bézier hadles will be calculated automatically.

Or you can use the grease pencil to draw and convert the strokes to curves.
Curve editing in blender is still quite rudimentary, for complex shapes it is a lot easier to use a vector drawing program (like inkscape, illustrator, etc), save the path as SVG and import into blender.